groupoid
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Article
28 sectionsContents
- Definitions
- Algebraic
- Category-theoretic
- Comparing the definitions
- Vertex groups and orbits
- Subgroupoids and morphisms
- Examples
- Fundamental groupoid
- Equivalence relation
- Examples
- Čech groupoid
- Group action
- Finite set
- Quotient variety
- Inertia groupoid
- Fiber product of groupoids
- Homological algebra
- Puzzles
- Mathieu groupoid
- Relation to groups
- Category of groupoids
- Relation to [[Category of small categories|Cat]]
- Relation to [[Simplicial set|sSet]]
- Groupoids in Grpd
- Groupoids with geometric structures
- See also
- Notes
- References
In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: Group with a partial function replacing the binary operation; Category in which every morphism is invertible. A category of this sort can be viewed as augmented with a unary operation on the morphisms, called inverse by analogy with group theory. A groupoid where there is only one object is a usual group.
In the presence of dependent typing, a category in general can be viewed as a typed monoid, and similarly, a groupoid can be viewed as simply a typed group. The morphisms take one from one object to another, and form a dependent family of types, thus morphisms might be typed , , say. Composition is then a total function: , so that .